Eisenstein-Kronecker series via the Poincar\'e bundle
Johannes Sprang

TL;DR
This paper offers a new algebraic approach to Eisenstein--Kronecker series and introduces a novel construction of Katz's $p$-adic Eisenstein measure using the Poincaré bundle and $p$-adic theta functions.
Contribution
It provides an alternative algebraic construction of Eisenstein--Kronecker series and a new conceptual framework for Katz's $p$-adic Eisenstein measure.
Findings
New algebraic construction of Eisenstein--Kronecker series via Poincaré bundle
A novel $p$-adic Eisenstein measure using $p$-adic theta functions
Enhanced understanding of algebraic and $p$-adic properties of Eisenstein series
Abstract
A classical construction of Katz gives a purely algebraic construction of Eisenstein--Kronecker series using the Gau\ss--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and -adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein--Kronecker series via the Poincar\'e bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable -adic Eisenstein measure through -adic theta functions of the Poincar\'e bundle.
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